Estimates of how many people demonstrated against Mohamed Morsi in Egypt at the end of June have varied by tens of millions of people. Is it really so hard to quantify such gigantic masses of human beings? The numbers are important because the size of the demonstrations was used to justify the military coup which followed them. If anyone had actually cared what these numbers were, the Egyptian military gave them a simple tool: aerial overhead videos of the demonstrations. Here is a reality check.

For a lot of Egyptians, the army’s move against the unpopular president and the Muslim Brothers was merely an exercise of the popular will, a new phase in the country’s revolution. They make wild claims about the number of demonstrators, declaring that there were more people calling for Morsi to step down than voted for him in the first place.

These claims seems to me to be ludicrously exaggerated, even if you ignore the conflation of the crowds’ calls for resignation with the military coup that followed, and even if you take as legitimate the idea of short-circuiting the electoral process via crowd action.

But let’s treat the idea of the “legitimacy of the crowd” seriously for a minute, nevertheless. Is there any possibility that the crowds in Tahrir really outweighed Morsi’s voters of last year?

Tahrir square, defined generously, is an irregular polygon taking in all of the empty space between the Mugamma and the Kentucky Fried Chicken franchise up near the corner of Talaat Harb Street, and from Qasr al-Aini street to the walls of the Arab Contractors construction site in front of the Egyptian Museum, about 160 meters by 250 meters. In addition, we should include open space in front of the Omar Makram mosque and all of the street surface area surrounding it, on Mohamed Mahmoud Street and Abd Al Qadir Hamza street; this adds another 60 by 120 meters of surface area to Tahrir Square, for a total of 47,200 square meters. Finally, we can throw in all 600 meters of Qasr al-Nil bridge, all the way from Tahrir and across the Nile to Opera Square, since there were frequently crowds on much or all of the bridge. At about 15 meters of pedestrian-usable width, that adds another nine thousand square meters. So we are looking at 56,200 square meters of space.

How many people can fit into a square meter? German scientific studies suggest that the maximum observed densities of large crowds are usually about 4 people per square meter, very rarely climbing above 5. Two people per square meter is the upper limit official guideline for most public events in the UK, and is a pretty dense crowd. Four people per square meter is the UK upper limit guideline for moving crowds (ie queues); 4.7 people per square meter is the maximum permitted crowd density at UK sporting events. I think we can safely take that as the upper limit for the overall density of the crowds at Tahrir two weeks ago. Assuming the crowds for the whole square and the bridge were packed equally densely, shoulder to shoulder, all the way to the Opera House, we are looking at a crowd of…264,140 people.

Typically crowds tend to thin out substantially toward the edges, meaning this is a very liberal estimate. I have here assumed 4.7 people per square meter; based on my view of the (Egyptian-Army provided) helicopter shots of the crowds, I think 2 people per square meter is a better estimate of average density, suggesting commensurately lower crowd figures.

There were simultaneous demonstrations in Heliopolis in front of the presidential palace. Let’s charitably say they packed the entire 45-meter wide Al Nadi Road, from Saleh Salem all the way to Mostafa Kemal intersection, a distance of over a kilometer. The videos of crowds I have seen at that demonstration did not appear to be more than 1 or 2 people per square meter (still a dense crowd), but let’s grant them 4.7 people per square meter as well, on the assumption that there was a point that they were packed like canned sardines, shoulder to shoulder for that entire kilometer. That gives us an additional crowd of 211,000 people. So in Cairo, the two biggest gatherings seem to me to have been of less, and possibly a lot less, than half a million people. Western news agency reports give vague figures for protests elsewhere, saying that ‘hundreds of thousands’ marched in Alexandria and other cities.

Morsi got nearly 3 million votes in Cairo and Giza, (and another 10 million elsewhere) in the June 2012 runoff election. The crowds two weeks ago were huge and exciting, but they were certainly a tiny fraction of the number of people who voted for Morsi. Even Morsi’s first-round votes, probably a better indication of his real support, (it subtracts the substantial anti-Shafiq vote of the second round), was well north of a million in Cairo and Giza.

The early sources for the crowd estimates of June 30…’millions’, seem to have been the military and the interior ministry (which remained under the military’s control despite the civilian government). Not terribly objective sources, since they were planning a coup based on the ‘legitimacy of the crowd.’

As time has gone by, estimates of the crowd sizes drifted into the realm of the improbable and then the absurd, born aloft, perhaps, by the opposition’s unconscious desire to justify the illegal seizure of power. We early on saw claims for 14 million demonstrators-a convenient number which is slightly more than the number of people who voted for Morsi last year. Later, a figure of 17 million was thrown around, suggesting that a crushing majority of last year’s voters were now on the street. Most recently, Nawal Sadawi has claimed that 34 million people demonstrated, another neat figure: a majority of the entire voting age population of Egypt. How convenient.

There is a much simpler way to play these numbers games, a way of establishing legitimacy that has worked well enough for 2,500 years of human history, and that is voting. Morsi should have been obliged to undergo an early referendum, perhaps timed to coincide with September’s parliamentary elections. A small deviation from constitutional norms, but a deviation I suspect he might have consented to. And given the impressive mobilization of the Egyptian people against his opaque and incompetent administration at the beginning of this month, that is a vote I suspect he might have lost.

What happened on June 30 must have been hallucinatory for Egypt’s liberal democracy activists: a vast uprising of the masses demanding that the state focus on bettering their lives, an uprising against a government of the Muslim Brothers. These are activists who have spent their lives in the shadow of the Muslim Brothers, openly belittled by Egypt’s mainstream Islamists as being out of touch with Egyptian society. As it turned out, it was the Brothers who were out of touch, pursuing a bland and ineffective social and economic agenda that conformed to their own social class, but not the Egyptian masses. Had there been true leadership among the opposition, this was a moment when a democratic alternative to the Brothers might well have emerged.

Tamarrud should have immediately and firmly rejected the army’s July 1 ‘ultimatum’ to Morsi. Instead, leaderless and disorganized, they fell into a well-laid trap, no doubt carefully planned by Al-Sisi and his grim little clique.

Jack Brown is co-editor of International Boulevard. He previously researched political repression in Egypt during the Mubarak regime.